How To Do Division

How to do Long Division

The problem we first looked at in short division, 478 divided by 6, we will do again using long division.

In long division our first step is the same as the short division, find the largest multiple of 6 which is less than or equal to 47.

We know is less than 47 so the first digit of our answer is 7. We write the 7 up over the 7 of 47. Then because we subtract 42 from the 47, which leaves 5.

We now bring down the 8 next to the 5 to give us 58.

We now need to find the largest multiple of 6 that is less than or equal to 58.

We write 9 up over the eight and we subtract 54 from the 58, leaving us with a remainder of 4 and we are finished.

Admittedly this is a rather short example but you can still see the difference and the similarity between the two methods of division. In both short and long division methods the first step is to work out what multiple of 6 is less than or equal to 47 then write the answer up above the 7 of 47. In short division you then do the subtraction in your head and write the remainder down next to the next digit, the 8. Whereas in long division you write out the subtraction and put the answer below then copy the 8 down next to the answer of the subtraction, the 5.

The next step for both division methods is to find how many times 6 goes into 58, which is 9 with a remainder of 4.

The calculations are the same in both methods but the short division is much faster than the long division as you are not writing everything down as you do in long division and you should arrive at the same answer regardless of the division method used.

We will have a look at another example, 36127 divided by 76.

Our first task is to find out how many times 76 goes into 361. This is where long division starts to lose friends, I mean who knows their 76 times tables. To find the answer we guess a number then multiply it by 76 to see if it is okay.
If we round 76 up to 80

This looks about right so we will try multiplying 76 by 4.

This looks okay so we put the 4 up up over the 1 of the 361 then we subtract 304 from 361 which leaves 57.

We bring the next digit in the dividend, the 2, down next to the 57 which gives us 572.

Now we need to find how many times 76 goes into 572.
Rounding 76 up to 80 we try:

Again this looks about right so we will multiply 76 by 7.

We write 7 up over the 2 in the dividend and then we subtract 532 from the 572.

We bring the next digit of the dividend, 7, down next to the 40 and we have 407.

We need to find how many times 76 goes into 407.
Estimating again by using 80 and multiply it by 5.

This is fairly close so we multiply 76 by 5.

We put the 5 above the 7 in the dividend then subtract 380 from 407.

We have our answer that 36127 divided by 76 is 475 with 27 remainder.

The problem with long division is that the numbers you are dealing with get beyond what most people are comfortable with particularly if trying to mentally do most of the calculations. Most people would have to scribble down the calculations on paper to work them out or as we did, round and estimate.
The next two methods of division try to overcome this limitation and keep the individual calculations required smaller and therefore easier to do mentally.

Comments

Since you picked two divisions that are difficult to do using the Vedic ‘crowning gem’ a.k.a. ‘flag’ method, it would be fair to do the same problems using the Trachtenberg system. That would be more convincing that the Trachtenberg method does not run into the same complications.